关于Diophantine方程x^2-Dy^2=±2的两个问题

Two Problems Concerning the Diophantine Equation x^2-Dy^2=±2

设D是不含素因数q≡5或7(mod 8)的无平方因子正奇数,p是大于3的奇素数.给出方程x^2-Dy^2=±2有正整数解(x,y)的充要条件,并且证明了当p≡17(mod 24)时,方程x^2-3py^2=-2必有正整数解(x,y).

Let D be a positive odd integer with square free and Dhas no primes q≡5 or 7（mod 8）,and let p be an odd prime with p3.In this paper,a necessary and sufficient condition for the equation x^2-Dy^2=±2 has positive integer solutions（x,y）is given.Moreover,we prove that if p≡17（mod 24）,then the equation x^2-Dy^2=-2 must have positive integer solutions（x,y）.